1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high-quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:17,400 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,400 --> 00:00:18,650 ocw.mit.edu. 8 00:00:55,550 --> 00:00:58,110 PROFESSOR: In the last lecture, I introduced and 9 00:00:58,110 --> 00:01:01,610 illustrated the kinds of signals and systems that we'll 10 00:01:01,610 --> 00:01:05,010 be dealing with throughout this course. 11 00:01:05,010 --> 00:01:08,820 In today's lecture I'd like to be a little more specific, and 12 00:01:08,820 --> 00:01:13,080 in particular, talk about some of the basic signals, both 13 00:01:13,080 --> 00:01:16,430 continuous-time and discrete-time that will form 14 00:01:16,430 --> 00:01:21,020 important building blocks as the course progresses. 15 00:01:21,020 --> 00:01:25,420 Let's begin with one signal, the continuous-time sinusoidal 16 00:01:25,420 --> 00:01:27,550 signal, which perhaps you're already 17 00:01:27,550 --> 00:01:29,650 somewhat familiar with. 18 00:01:29,650 --> 00:01:32,790 Mathematically, the continuous-time sinusoidal 19 00:01:32,790 --> 00:01:36,500 signal is expressed as I've indicated here. 20 00:01:36,500 --> 00:01:41,750 There are three parameters, A, omega_0 and phi. 21 00:01:41,750 --> 00:01:46,760 The parameter A is referred to as the amplitude, the 22 00:01:46,760 --> 00:01:51,150 parameter omega 0 as the frequency, and the parameter 23 00:01:51,150 --> 00:01:54,950 phi as the phase. 24 00:01:54,950 --> 00:02:00,010 And graphically, the continuous-time sinusoidal 25 00:02:00,010 --> 00:02:04,440 signal has the form shown here. 26 00:02:04,440 --> 00:02:08,759 Now, the sinusoidal signal has a number of important 27 00:02:08,759 --> 00:02:13,160 properties that we'll find it convenient to exploit as the 28 00:02:13,160 --> 00:02:18,800 course goes along, one of which is the fact that the 29 00:02:18,800 --> 00:02:24,250 sinusoidal signal is what is referred to as periodic. 30 00:02:24,250 --> 00:02:28,920 What I mean by periodic is that under an appropriate time 31 00:02:28,920 --> 00:02:34,380 shift, which I indicate here as T_0, the signal replicates 32 00:02:34,380 --> 00:02:36,110 or repeats itself. 33 00:02:36,110 --> 00:02:40,820 Or said another way, if we shift the time origin by an 34 00:02:40,820 --> 00:02:44,690 appropriate amount T_0, the smallest value T_0 being 35 00:02:44,690 --> 00:02:48,190 referred to as the period, then x(t) is 36 00:02:48,190 --> 00:02:51,420 equal to itself, shifted. 37 00:02:51,420 --> 00:02:57,630 And we can demonstrate it mathematically by simply 38 00:02:57,630 --> 00:03:00,700 substituting into the mathematical expression for 39 00:03:00,700 --> 00:03:07,630 the sinusoidal signal t + T_0, in place of t. 40 00:03:07,630 --> 00:03:11,310 When we carry out the expansion we then have, for 41 00:03:11,310 --> 00:03:20,040 the argument of the sinusoid, omega_0 t + omega_0 T_0 + phi. 42 00:03:20,040 --> 00:03:23,350 Now, one of the things that we know about sinusoidal 43 00:03:23,350 --> 00:03:27,080 functions is that if you change the argument by any 44 00:03:27,080 --> 00:03:30,630 integer multiple of 2 pi, then the function 45 00:03:30,630 --> 00:03:32,320 has the same value. 46 00:03:32,320 --> 00:03:36,110 And so we can exploit that here, in particular with 47 00:03:36,110 --> 00:03:41,400 omega_0 T_0 and integer multiple of 2 pi. 48 00:03:41,400 --> 00:03:44,700 Then the right-hand side of this equation is equal to the 49 00:03:44,700 --> 00:03:47,040 left-hand side of the equation. 50 00:03:47,040 --> 00:03:54,880 So with omega_0 T_0 equal to 2 pi times an integer, or T_0 51 00:03:54,880 --> 00:03:57,420 equal to 2 pi times an integer divided by 52 00:03:57,420 --> 00:04:00,280 omega_0, the signal repeats. 53 00:04:00,280 --> 00:04:03,700 The period is defined as the smallest value of T_0. 54 00:04:03,700 --> 00:04:08,750 And so the period is 2 pi divided by omega_0. 55 00:04:08,750 --> 00:04:11,190 And going back to our sinusoidal 56 00:04:11,190 --> 00:04:16,160 signal, we can see that-- 57 00:04:16,160 --> 00:04:23,100 and I've indicated here, then, the period as 2 pi / omega_0. 58 00:04:23,100 --> 00:04:26,990 And that's the value under which the signal repeats. 59 00:04:30,160 --> 00:04:35,710 Now in addition, a useful property of the sinusoidal 60 00:04:35,710 --> 00:04:42,070 signal is the fact that a time shift of a sinusoid is 61 00:04:42,070 --> 00:04:44,290 equivalent to a phase change. 62 00:04:44,290 --> 00:04:48,670 And we can demonstrate that again mathematically, in 63 00:04:48,670 --> 00:04:52,610 particular if we put the sinusoidal signal 64 00:04:52,610 --> 00:04:54,540 under a time shift-- 65 00:04:54,540 --> 00:04:57,710 I've indicated the time shift that I'm 66 00:04:57,710 --> 00:05:00,680 talking about by t_0-- 67 00:05:00,680 --> 00:05:06,570 and expand this out, then we see that that is equivalent to 68 00:05:06,570 --> 00:05:08,760 a change in phase. 69 00:05:08,760 --> 00:05:12,640 And an important thing to recognize about this statement 70 00:05:12,640 --> 00:05:18,260 is that not only is a time shift generating a phase 71 00:05:18,260 --> 00:05:24,890 change, but, in fact, if we inserted a phase change, there 72 00:05:24,890 --> 00:05:31,570 is always a value of t_0 which would correspond to an 73 00:05:31,570 --> 00:05:33,420 equivalent time shift. 74 00:05:33,420 --> 00:05:39,630 Said another way, if we take omega_0 t_0 and think of that 75 00:05:39,630 --> 00:05:45,190 as our change in phase, for any change in phase, we can 76 00:05:45,190 --> 00:05:48,900 solve this equation for a time shift, or conversely for any 77 00:05:48,900 --> 00:05:51,810 value of time shift, that represents 78 00:05:51,810 --> 00:05:54,740 an appropriate phase. 79 00:05:54,740 --> 00:05:58,820 So a time shift corresponds to a phase change, and a phase 80 00:05:58,820 --> 00:06:02,970 change, likewise, corresponds to time shift. 81 00:06:02,970 --> 00:06:11,160 And so for example, if we look at the general sinusoidal 82 00:06:11,160 --> 00:06:17,530 signal that we saw previously, in effect, changing the phase 83 00:06:17,530 --> 00:06:21,090 corresponds to moving this signal in time 84 00:06:21,090 --> 00:06:22,990 one way or the other. 85 00:06:22,990 --> 00:06:27,110 For example, if we look at the sinusoidal signal with a phase 86 00:06:27,110 --> 00:06:31,820 equal to 0 that corresponds to locating the time 87 00:06:31,820 --> 00:06:33,950 origin at this peak. 88 00:06:33,950 --> 00:06:41,820 And I've indicated that on the following graph. 89 00:06:41,820 --> 00:06:46,980 So here we have illustrated a sinusoid with 0 phase, or a 90 00:06:46,980 --> 00:06:50,650 cosine with 0 phase, corresponding to taking our 91 00:06:50,650 --> 00:06:52,890 general picture and shifting it. 92 00:06:52,890 --> 00:06:56,730 Shifting it appropriately as I've indicated here. 93 00:06:56,730 --> 00:07:00,370 This, of course, still has the property that it's a periodic 94 00:07:00,370 --> 00:07:05,520 function, since we simply displaced it in time. 95 00:07:05,520 --> 00:07:09,580 And by looking at the graph, what we see is that it has 96 00:07:09,580 --> 00:07:12,540 another very important property, a property referred 97 00:07:12,540 --> 00:07:14,070 to as even. 98 00:07:14,070 --> 00:07:18,180 And that's a property that we'll find useful, in general, 99 00:07:18,180 --> 00:07:20,970 to refer to in relation to signals. 100 00:07:20,970 --> 00:07:24,860 A signal is said to be even if, when we reflect it about 101 00:07:24,860 --> 00:07:27,170 the origin, it looks exactly the same. 102 00:07:27,170 --> 00:07:30,600 So it's symmetric about the origin. 103 00:07:30,600 --> 00:07:34,670 And looking at this sinusoid, that, in 104 00:07:34,670 --> 00:07:37,010 fact, has that property. 105 00:07:37,010 --> 00:07:41,550 And mathematically, the statement that it's even is 106 00:07:41,550 --> 00:07:45,570 equivalent to the statement that if we replace the time 107 00:07:45,570 --> 00:07:48,630 argument by its negative, the function 108 00:07:48,630 --> 00:07:50,070 itself doesn't change. 109 00:07:53,290 --> 00:07:56,660 Now this corresponded to a phase shift of 0 in our 110 00:07:56,660 --> 00:07:58,990 original cosine expression. 111 00:07:58,990 --> 00:08:03,350 If instead, we had chosen a phase shift of, let's say, 112 00:08:03,350 --> 00:08:08,275 -pi/2, then instead of a cosinusoidal signal, what we 113 00:08:08,275 --> 00:08:14,770 would regenerate is a sinusoid with the appropriate phase. 114 00:08:14,770 --> 00:08:22,260 Or, said another way, if we take our original cosine and 115 00:08:22,260 --> 00:08:27,120 substitute in for the phase -pi/2, then of course we have 116 00:08:27,120 --> 00:08:29,540 this mathematical expression. 117 00:08:29,540 --> 00:08:32,900 Using just straightforward trigonometric identities, we 118 00:08:32,900 --> 00:08:37,799 can express that alternately as sin(omega_0*t). 119 00:08:37,799 --> 00:08:40,990 The frequency and amplitude, of course, haven't changed. 120 00:08:40,990 --> 00:08:45,750 And that, you can convince yourself, also is equivalent 121 00:08:45,750 --> 00:08:50,500 to shifting the cosine by an amount in time that I've 122 00:08:50,500 --> 00:08:54,600 indicated here, namely a quarter of a period. 123 00:08:54,600 --> 00:09:00,160 So illustrated below is the graph now, when we have a 124 00:09:00,160 --> 00:09:07,290 phase of -pi/2 in our cosine, which is a sinusoidal signal. 125 00:09:07,290 --> 00:09:09,270 Of course, it's still periodic. 126 00:09:09,270 --> 00:09:15,220 It's periodic with a period of 2 pi / omega_0 again, because 127 00:09:15,220 --> 00:09:19,690 all that we've done by introducing a phase change is 128 00:09:19,690 --> 00:09:20,940 introduced the time shift. 129 00:09:23,780 --> 00:09:27,420 Now, when we look at the sinusoid in comparison with 130 00:09:27,420 --> 00:09:30,730 the cosine, namely with this particular choice of phase, 131 00:09:30,730 --> 00:09:34,960 this has a different symmetry, and that symmetry 132 00:09:34,960 --> 00:09:37,020 is referred to odd. 133 00:09:37,020 --> 00:09:41,590 What odd symmetry means, graphically, is that when we 134 00:09:41,590 --> 00:09:50,230 flip the signal about the time origin, we also multiply it by 135 00:09:50,230 --> 00:09:51,410 a minus sign. 136 00:09:51,410 --> 00:09:55,110 So that's, in effect, anti-symmetric. 137 00:09:55,110 --> 00:09:56,880 It's not the mirror image, but it's the mirror 138 00:09:56,880 --> 00:09:58,390 image flipped over. 139 00:09:58,390 --> 00:10:02,420 And we'll find many occasions, not only to refer to signals 140 00:10:02,420 --> 00:10:06,610 more general than sinusoidal signals, as even in some cases 141 00:10:06,610 --> 00:10:08,950 and odd in other cases. 142 00:10:08,950 --> 00:10:14,240 And in general, mathematically, an odd signal 143 00:10:14,240 --> 00:10:22,190 is one which satisfies the algebraic expression, x(t). 144 00:10:22,190 --> 00:10:29,660 When you replace t by its negative, is equal to -x(-t). 145 00:10:29,660 --> 00:10:35,640 So replacing the argument by its negative corresponds to an 146 00:10:35,640 --> 00:10:36,910 algebraic sign reversal. 147 00:10:39,570 --> 00:10:39,900 OK. 148 00:10:39,900 --> 00:10:43,610 So this is the class of continuous-time sinusoids. 149 00:10:43,610 --> 00:10:46,470 We'll have a little more to say about it later. 150 00:10:46,470 --> 00:10:51,720 But I'd now like to turn to discrete-time sinusoids. 151 00:10:51,720 --> 00:10:54,790 What we'll see is that discrete-time sinusoids are 152 00:10:54,790 --> 00:10:59,070 very much like continuous-time ones, but also with some very 153 00:10:59,070 --> 00:11:00,650 important differences. 154 00:11:00,650 --> 00:11:03,920 And we want to focus, not only on the similarities, but also 155 00:11:03,920 --> 00:11:06,040 on the differences. 156 00:11:06,040 --> 00:11:10,530 Well, let's begin with the mathematical expression. 157 00:11:10,530 --> 00:11:16,740 A discrete-time sinusoidal signal, mathematically, is as 158 00:11:16,740 --> 00:11:22,040 I've indicated here, A cos(omega_0 n + phi). 159 00:11:22,040 --> 00:11:26,050 And just as in the continuous-time case, the 160 00:11:26,050 --> 00:11:31,360 parameter A is what we'll refer to as the amplitude, 161 00:11:31,360 --> 00:11:39,820 omega_0 as the frequency, and phi as the phase. 162 00:11:39,820 --> 00:11:46,420 And I've illustrated here several discrete-time 163 00:11:46,420 --> 00:11:48,480 sinusoidal signals. 164 00:11:48,480 --> 00:11:51,000 And they kind of look similar. 165 00:11:51,000 --> 00:11:53,720 In fact, if you track what you might think of as the 166 00:11:53,720 --> 00:11:57,670 envelope, it looks very much like what a continuous-time 167 00:11:57,670 --> 00:12:00,090 sinusoid might look like. 168 00:12:00,090 --> 00:12:03,720 But keep in mind that the independent variable, in this 169 00:12:03,720 --> 00:12:06,890 case, is an integer variable. 170 00:12:06,890 --> 00:12:13,050 And so the sequence only takes on values at integer values of 171 00:12:13,050 --> 00:12:13,810 the argument. 172 00:12:13,810 --> 00:12:19,540 And we'll see that has a very important implication, and 173 00:12:19,540 --> 00:12:22,000 we'll see that shortly. 174 00:12:22,000 --> 00:12:24,950 Now, one of the issues that we addressed in the 175 00:12:24,950 --> 00:12:27,500 continuous-time case was periodicity. 176 00:12:27,500 --> 00:12:33,150 And I want to return to that shortly, because that is one 177 00:12:33,150 --> 00:12:35,085 of the areas where there is an important distinction. 178 00:12:37,770 --> 00:12:42,880 Let's first, though, examine the statement similar to the 179 00:12:42,880 --> 00:12:46,170 one that we examined for continuous time, namely the 180 00:12:46,170 --> 00:12:51,730 relationship between a time shift and a phase change. 181 00:12:51,730 --> 00:12:56,490 Now, in continuous time, of course, we saw that a time 182 00:12:56,490 --> 00:13:00,360 shift corresponds to a phase change, and vice versa. 183 00:13:00,360 --> 00:13:04,460 Let's first look at the relationship between shifting 184 00:13:04,460 --> 00:13:07,470 time and generating a change in phase. 185 00:13:07,470 --> 00:13:12,400 In particular for discrete time, if I implement a time 186 00:13:12,400 --> 00:13:16,480 shift that generates a phase change-- 187 00:13:16,480 --> 00:13:21,160 and we can see that easily by simply inserting a 188 00:13:21,160 --> 00:13:24,300 time shift, n + n_0. 189 00:13:24,300 --> 00:13:29,860 And if we expand out this argument, we have omega_0 n + 190 00:13:29,860 --> 00:13:31,670 omega_0 n_0. 191 00:13:31,670 --> 00:13:34,300 And so I've done that on the right-hand side of the 192 00:13:34,300 --> 00:13:36,900 equation here. 193 00:13:36,900 --> 00:13:41,300 And the omega_0 n_0, then, simply corresponds 194 00:13:41,300 --> 00:13:44,360 to a change in phase. 195 00:13:44,360 --> 00:13:51,390 So clearly, a shift in time generates a change in phase. 196 00:13:51,390 --> 00:13:57,650 And for example, if we take a particular sinusoidal signal, 197 00:13:57,650 --> 00:14:01,050 let's say we take the cosine signal at a particular 198 00:14:01,050 --> 00:14:06,830 frequency, and with a phase equal to 0, a sequence that we 199 00:14:06,830 --> 00:14:12,370 might generate is one that I've illustrated here. 200 00:14:12,370 --> 00:14:16,510 So what I'm illustrating here is the cosine 201 00:14:16,510 --> 00:14:20,230 signal with 0 phase. 202 00:14:20,230 --> 00:14:24,570 And it has a particular behavior to it, which will 203 00:14:24,570 --> 00:14:27,600 depend somewhat on the frequency. 204 00:14:27,600 --> 00:14:34,310 If I now take this same sequence and shift it so that 205 00:14:34,310 --> 00:14:39,230 the time origin is shifted a quarter of a period away, then 206 00:14:39,230 --> 00:14:41,140 you can convince yourself-- 207 00:14:41,140 --> 00:14:43,780 and it's straightforward to work out-- that that time 208 00:14:43,780 --> 00:14:49,030 shift corresponds to a phase shift of pi/2. 209 00:14:49,030 --> 00:14:57,580 So in that case, with the cosine with a phase of -pi/2, 210 00:14:57,580 --> 00:15:05,410 that will correspond to the expression that I have here. 211 00:15:05,410 --> 00:15:07,900 We could alternately write that, using again a 212 00:15:07,900 --> 00:15:12,080 trigonometric identity, as a sine function. 213 00:15:12,080 --> 00:15:17,790 And that, I've stated, is equivalent to a time shift. 214 00:15:17,790 --> 00:15:23,360 Namely, this shift of pi/2 is equal to a certain time shift, 215 00:15:23,360 --> 00:15:27,470 and the time shift for this particular example is a 216 00:15:27,470 --> 00:15:29,560 quarter of a period. 217 00:15:29,560 --> 00:15:35,210 So here, we have the sinusoid. 218 00:15:35,210 --> 00:15:37,700 Previously we had the cosine. 219 00:15:37,700 --> 00:15:40,590 The cosine was exactly the same sequence, but with the 220 00:15:40,590 --> 00:15:42,520 origin located here. 221 00:15:42,520 --> 00:15:45,310 And in fact, that's exactly the way we drew this graph. 222 00:15:45,310 --> 00:15:49,120 Namely, we just simply took the same values and changed 223 00:15:49,120 --> 00:15:51,330 the time origin. 224 00:15:51,330 --> 00:15:56,070 Now, looking at this sequence, which is the sinusoidal 225 00:15:56,070 --> 00:16:01,120 sequence, the phase of -pi/2, that has a certain symmetry. 226 00:16:01,120 --> 00:16:05,440 And in fact, what we see is that it has an odd symmetry, 227 00:16:05,440 --> 00:16:07,790 just as in the continuous-time case. 228 00:16:07,790 --> 00:16:11,190 Namely, if we take that sequence, flip it about the 229 00:16:11,190 --> 00:16:15,080 axis, and flip it over in sign, that we get the same 230 00:16:15,080 --> 00:16:17,380 sequence back again. 231 00:16:17,380 --> 00:16:22,050 Whereas with 0 phase corresponding to the cosine 232 00:16:22,050 --> 00:16:26,330 that I showed previously, that has an even symmetry. 233 00:16:26,330 --> 00:16:31,070 Namely, if I flip it about the time origin and don't do a 234 00:16:31,070 --> 00:16:34,650 sign reversal, then the sequence is maintained. 235 00:16:34,650 --> 00:16:40,650 So here, we have an odd symmetry, expressed 236 00:16:40,650 --> 00:16:43,840 mathematically as I've indicated. 237 00:16:43,840 --> 00:16:46,980 Namely, replacing the independent variable by its 238 00:16:46,980 --> 00:16:52,370 negative attaches a negative sign to the whole sequence. 239 00:16:52,370 --> 00:17:00,660 Whereas in the previous case, what we have is 0 phase and an 240 00:17:00,660 --> 00:17:02,230 even symmetry. 241 00:17:02,230 --> 00:17:06,240 And that's expressed mathematically as x[n] 242 00:17:06,240 --> 00:17:07,490 = x[-n]. 243 00:17:11,839 --> 00:17:14,900 Now, one of the things I've said so far about 244 00:17:14,900 --> 00:17:19,849 discrete-time sinusoids is that a time shift corresponds 245 00:17:19,849 --> 00:17:22,140 to a phase change. 246 00:17:22,140 --> 00:17:26,960 And we can then ask whether the reverse statement is also 247 00:17:26,960 --> 00:17:29,950 true, and we knew that the reverse statement was true in 248 00:17:29,950 --> 00:17:32,160 continuous time. 249 00:17:32,160 --> 00:17:38,920 Specifically, is it true that a phase change always 250 00:17:38,920 --> 00:17:40,580 corresponds to a time shift? 251 00:17:40,580 --> 00:17:43,100 Now, we know that that is true, namely, that this 252 00:17:43,100 --> 00:17:47,160 statement works both ways in continuous time. 253 00:17:47,160 --> 00:17:49,300 Does it in discrete time? 254 00:17:49,300 --> 00:17:54,040 Well, the answer, somewhat interestingly or surprisingly 255 00:17:54,040 --> 00:17:58,480 until you sit down and think about it, is no. 256 00:17:58,480 --> 00:18:02,460 It is not necessarily true in discrete time that any phase 257 00:18:02,460 --> 00:18:06,830 change can be interpreted as a simple time 258 00:18:06,830 --> 00:18:08,430 shift of the sequence. 259 00:18:08,430 --> 00:18:12,210 And let me just indicate what the problem is. 260 00:18:12,210 --> 00:18:17,880 If we look at the relationship between the left side and the 261 00:18:17,880 --> 00:18:21,660 right side of this equation, expanding this out as we did 262 00:18:21,660 --> 00:18:27,950 previously, we have that omega_0 n + omega_0 n_0 must 263 00:18:27,950 --> 00:18:31,370 correspond to omega_0 n + phi. 264 00:18:31,370 --> 00:18:35,200 And so omega_0 n_0 must correspond 265 00:18:35,200 --> 00:18:37,630 to the phase change. 266 00:18:37,630 --> 00:18:43,200 Now, what you can see pretty clearly is that depending on 267 00:18:43,200 --> 00:18:48,730 the relationship between phi and omega_0, n_0 may or may 268 00:18:48,730 --> 00:18:51,950 not come out to be an integer. 269 00:18:51,950 --> 00:18:56,400 Now, in continuous time, the amount of time shift did not 270 00:18:56,400 --> 00:18:58,350 have to be an integer amount. 271 00:18:58,350 --> 00:19:01,520 In discrete time, when we talk about a time shift, the amount 272 00:19:01,520 --> 00:19:03,760 of time shift-- obviously, because of the nature of 273 00:19:03,760 --> 00:19:05,270 discrete time signals-- 274 00:19:05,270 --> 00:19:06,640 must be an integer. 275 00:19:06,640 --> 00:19:12,570 So the phase changes related to time shifts must satisfy 276 00:19:12,570 --> 00:19:14,580 this particular relationship. 277 00:19:14,580 --> 00:19:19,380 Namely, that omega_0 n_0, where n_0 is an integer, is 278 00:19:19,380 --> 00:19:21,145 equal to the change in phase. 279 00:19:24,940 --> 00:19:25,480 OK. 280 00:19:25,480 --> 00:19:29,350 Now, that's one distinction between continuous time and 281 00:19:29,350 --> 00:19:30,290 discrete time. 282 00:19:30,290 --> 00:19:34,530 Let's now focus on another one, namely the issue of 283 00:19:34,530 --> 00:19:36,190 periodicity. 284 00:19:36,190 --> 00:19:39,930 And what we'll see is that again, whereas in continuous 285 00:19:39,930 --> 00:19:44,930 time, all continuous-time sinusoids are periodic, in the 286 00:19:44,930 --> 00:19:48,220 discrete-time case that is not necessarily true. 287 00:19:51,480 --> 00:19:56,960 To explore that a little more carefully, let's look at the 288 00:19:56,960 --> 00:20:00,750 expression, again, for a general sinusoidal signal with 289 00:20:00,750 --> 00:20:05,520 an arbitrary amplitude, frequency, and phase. 290 00:20:05,520 --> 00:20:10,590 And for this to be periodic, what we require is that there 291 00:20:10,590 --> 00:20:15,850 be some value, N, under which, when we shift the sequence by 292 00:20:15,850 --> 00:20:19,150 that amount, we get the same sequence back again. 293 00:20:19,150 --> 00:20:22,000 And the smallest-value N is what we've 294 00:20:22,000 --> 00:20:24,050 defined as the period. 295 00:20:24,050 --> 00:20:27,870 Now, when we try that on a sinusoid, we of course 296 00:20:27,870 --> 00:20:35,460 substitute in for n, n + N. And when we expand out the 297 00:20:35,460 --> 00:20:41,910 argument here, we'll get the argument that I have on the 298 00:20:41,910 --> 00:20:44,210 right-hand side. 299 00:20:44,210 --> 00:20:48,960 And in order for this to repeat, in other words, in 300 00:20:48,960 --> 00:20:55,630 order for us to discard this term, omega_0 N, where N is 301 00:20:55,630 --> 00:21:00,100 the period, must be an integer multiple of 2 pi. 302 00:21:00,100 --> 00:21:06,720 And in that case, it's periodic as long as omega_0 N, 303 00:21:06,720 --> 00:21:11,550 N being the period, is 2 pi times an integer. 304 00:21:11,550 --> 00:21:15,550 Just simply dividing this out, we have N, the period, is 2 pi 305 00:21:15,550 --> 00:21:20,900 m / omega_0. 306 00:21:20,900 --> 00:21:23,880 Well, you could say, OK what's the big deal? 307 00:21:23,880 --> 00:21:26,690 Whatever N happens to come out to be when we do that little 308 00:21:26,690 --> 00:21:29,010 bit of algebra, that's the period. 309 00:21:29,010 --> 00:21:35,910 But in fact, N, or 2 pi m / omega_0, may not ever come out 310 00:21:35,910 --> 00:21:37,270 to be an integer. 311 00:21:37,270 --> 00:21:38,970 Or it may not come out to be the one that 312 00:21:38,970 --> 00:21:39,850 you thought it might. 313 00:21:39,850 --> 00:21:43,340 For example, let's look at some 314 00:21:43,340 --> 00:21:45,895 particular sinusoidal signals. 315 00:21:49,050 --> 00:21:49,430 Let's see. 316 00:21:49,430 --> 00:21:53,160 We have the first one here, which is a 317 00:21:53,160 --> 00:21:55,880 sinusoid, as I've shown. 318 00:21:55,880 --> 00:21:59,110 And it has a frequency, what I've referred to as the 319 00:21:59,110 --> 00:22:05,130 frequency, omega_0 = 2 pi / 12. 320 00:22:05,130 --> 00:22:11,720 And what we'd like to look at is 2 pi / omega_0, then find 321 00:22:11,720 --> 00:22:14,070 an integer to multiply that by in order 322 00:22:14,070 --> 00:22:15,740 to get another integer. 323 00:22:15,740 --> 00:22:17,110 Let's just try that here. 324 00:22:17,110 --> 00:22:24,760 If we look at 2 pi / omega_0, 2 pi / omega_0, for this case, 325 00:22:24,760 --> 00:22:27,130 is equal to 12. 326 00:22:27,130 --> 00:22:28,530 Well, that's fine. 327 00:22:28,530 --> 00:22:29,890 12 is an integer. 328 00:22:29,890 --> 00:22:35,030 So what that says is that this sinusoidal signal is periodic. 329 00:22:35,030 --> 00:22:39,990 And in fact, it's periodic with a period of 12. 330 00:22:39,990 --> 00:22:41,820 Let's look at the next one. 331 00:22:41,820 --> 00:22:46,810 The next one, we would have 2 pi / omega_0 again. 332 00:22:46,810 --> 00:22:48,060 And that's equal to 31/4. 333 00:22:50,560 --> 00:22:57,080 So what that says is that the period is 31/4. 334 00:22:57,080 --> 00:22:58,440 But wait a minute. 335 00:22:58,440 --> 00:23:00,730 31/4 isn't an integer. 336 00:23:00,730 --> 00:23:02,820 We have to multiply that by an integer 337 00:23:02,820 --> 00:23:04,730 to get another integer. 338 00:23:04,730 --> 00:23:09,680 Well, we'd multiply that by 4, so (2 pi / omega_0) times 4 is 339 00:23:09,680 --> 00:23:11,790 31, 31 is an integer. 340 00:23:11,790 --> 00:23:16,290 And so what that says is this is periodic, not with a period 341 00:23:16,290 --> 00:23:22,130 of 2 pi / omega_0, but with a period of (2 pi / omega_0) 342 00:23:22,130 --> 00:23:29,950 times 4, namely with a period of 31. 343 00:23:29,950 --> 00:23:34,310 Finally, let's take the example where omega_0 is equal 344 00:23:34,310 --> 00:23:37,030 to 1/6, as I've shown here. 345 00:23:37,030 --> 00:23:40,330 That actually looks, if you track it with your eye, like 346 00:23:40,330 --> 00:23:41,970 it's periodic. 347 00:23:41,970 --> 00:23:47,660 2 pi / omega_0, in that case, is equal to 12 pi. 348 00:23:47,660 --> 00:23:53,070 Well, what integer can I multiply 12 pi by and get 349 00:23:53,070 --> 00:23:54,400 another integer? 350 00:23:54,400 --> 00:23:58,350 The answer is none, because pi is an irrational number. 351 00:23:58,350 --> 00:24:03,860 So in fact, what that says is that if you look at this 352 00:24:03,860 --> 00:24:09,380 sinusoidal signal, it's not periodic at all, even though 353 00:24:09,380 --> 00:24:13,120 you might fool yourself into thinking it is simply because 354 00:24:13,120 --> 00:24:15,830 the envelope looks periodic. 355 00:24:15,830 --> 00:24:19,510 Namely, the continuous-time equivalent of this is 356 00:24:19,510 --> 00:24:22,580 periodic, the discrete-time sequence is not. 357 00:24:26,750 --> 00:24:27,180 OK. 358 00:24:27,180 --> 00:24:31,860 Well, we've seen, then, some important distinctions between 359 00:24:31,860 --> 00:24:34,780 continuous-time sinusoidal signals and discrete-time 360 00:24:34,780 --> 00:24:36,730 sinusoidal signals. 361 00:24:36,730 --> 00:24:41,900 The first one is the fact that in the continuous-time case, a 362 00:24:41,900 --> 00:24:45,830 time shift and phase change are always equivalent. 363 00:24:45,830 --> 00:24:49,210 Whereas in the discrete-time case, in effect, it works one 364 00:24:49,210 --> 00:24:52,400 way but not the other way. 365 00:24:52,400 --> 00:24:58,060 We've also seen that for a continuous-time signal, the 366 00:24:58,060 --> 00:25:01,990 continuous-time signal is always periodic, whereas the 367 00:25:01,990 --> 00:25:04,850 discrete-time signal is not necessarily. 368 00:25:04,850 --> 00:25:08,640 In particular, for the continuous-time case, if we 369 00:25:08,640 --> 00:25:12,320 have a general expression for the sinusoidal signal that 370 00:25:12,320 --> 00:25:15,650 I've indicated here, that's periodic for 371 00:25:15,650 --> 00:25:18,440 any choice of omega_0. 372 00:25:18,440 --> 00:25:30,880 Whereas in the discrete-time case, it's periodic only if 2 373 00:25:30,880 --> 00:25:34,740 pi / omega_0 can be multiplied by an integer 374 00:25:34,740 --> 00:25:37,020 to get another integer. 375 00:25:37,020 --> 00:25:41,470 Now, another important and, as it turns out, useful 376 00:25:41,470 --> 00:25:44,940 distinction between the continuous-time and 377 00:25:44,940 --> 00:25:49,370 discrete-time case is the fact that in the discrete-time 378 00:25:49,370 --> 00:25:56,550 case, as we vary what I've called the frequency omega_0, 379 00:25:56,550 --> 00:26:02,390 we only see distinct signals as omega_0 380 00:26:02,390 --> 00:26:04,900 varies over a 2 pi interval. 381 00:26:04,900 --> 00:26:09,940 And if we let omega_0 vary outside the range of, let's 382 00:26:09,940 --> 00:26:15,130 say, -pi to pi, or 0 to 2 pi, we'll see the same sequences 383 00:26:15,130 --> 00:26:18,870 all over again, even though at first glance, the mathematical 384 00:26:18,870 --> 00:26:20,870 expression might look different. 385 00:26:20,870 --> 00:26:26,980 So in the discrete-time case, this class of signals is 386 00:26:26,980 --> 00:26:33,340 identical for values of omega_0 separated by 2 pi, 387 00:26:33,340 --> 00:26:37,890 whereas in the continuous-time case, that is not true. 388 00:26:37,890 --> 00:26:41,020 In particular, if I consider these sinusoidal 389 00:26:41,020 --> 00:26:48,260 continuous-time signals, as I vary omega_0, what will happen 390 00:26:48,260 --> 00:26:53,200 is that I will always see different sinusoidal signals. 391 00:26:53,200 --> 00:26:55,210 Namely, these won't be equal. 392 00:26:55,210 --> 00:27:00,970 And in effect, we can justify that statement algebraically. 393 00:27:00,970 --> 00:27:04,410 And I won't take the time to do it carefully. 394 00:27:04,410 --> 00:27:09,670 But let's look, first of all, at the discrete-time case. 395 00:27:09,670 --> 00:27:15,130 And the statement that I'm making is that if I have two 396 00:27:15,130 --> 00:27:18,700 discrete-time sinusoidal signals at two different 397 00:27:18,700 --> 00:27:24,980 frequencies, and if these frequencies are separated by 398 00:27:24,980 --> 00:27:30,170 an integer multiple of 2 pi-- namely if omega_2 is equal to 399 00:27:30,170 --> 00:27:35,650 omega_1 + 2 pi times an integer m-- 400 00:27:35,650 --> 00:27:40,070 when I substitute this into this expression, because of 401 00:27:40,070 --> 00:27:46,330 the fact that n is also an integer, I'll have m * n as an 402 00:27:46,330 --> 00:27:48,270 integer multiple of 2 pi. 403 00:27:48,270 --> 00:27:50,520 And that term, of course, will disappear because of the 404 00:27:50,520 --> 00:27:53,370 periodicity of the sinusoid, and these two 405 00:27:53,370 --> 00:27:56,350 sequences will be equal. 406 00:27:56,350 --> 00:28:03,450 On the other hand in the continuous-time case, since t 407 00:28:03,450 --> 00:28:06,560 is not restricted to be an integer variable, for 408 00:28:06,560 --> 00:28:11,230 different values of omega_1 and omega_2, these sinusoidal 409 00:28:11,230 --> 00:28:13,770 signals will always be different. 410 00:28:17,280 --> 00:28:17,650 OK. 411 00:28:17,650 --> 00:28:21,540 Now, many of the issues that I've raised so far, in 412 00:28:21,540 --> 00:28:24,550 relation to sinusoidal signals, are elaborated on in 413 00:28:24,550 --> 00:28:26,240 more detail in the text. 414 00:28:26,240 --> 00:28:30,890 And of course, you'll have an opportunity to exercise some 415 00:28:30,890 --> 00:28:36,150 of this as you work through the video course manual. 416 00:28:36,150 --> 00:28:40,550 Let me stress that sinusoidal signals will play an extremely 417 00:28:40,550 --> 00:28:44,260 important role for us as building blocks for general 418 00:28:44,260 --> 00:28:47,770 signals and descriptions of systems, and leads to the 419 00:28:47,770 --> 00:28:51,870 whole concept Fourier analysis, which is very 420 00:28:51,870 --> 00:28:53,755 heavily exploited throughout the course. 421 00:28:56,760 --> 00:29:02,870 What I'd now like to turn to is another class of important 422 00:29:02,870 --> 00:29:03,780 building blocks. 423 00:29:03,780 --> 00:29:06,820 And in fact, we'll see that under certain conditions, 424 00:29:06,820 --> 00:29:10,970 these relate strongly to sinusoidal signals, namely the 425 00:29:10,970 --> 00:29:14,580 class of real and complex exponentials. 426 00:29:14,580 --> 00:29:19,810 Let me begin, first of all, with the real exponential, and 427 00:29:19,810 --> 00:29:23,420 in particular, in the continuous-time case. 428 00:29:23,420 --> 00:29:26,980 A real continuous-time exponential is mathematically 429 00:29:26,980 --> 00:29:36,000 expressed, as I indicate here, x(t) = C e ^ (a t), where for 430 00:29:36,000 --> 00:29:40,320 the real exponential, C and a are real numbers. 431 00:29:40,320 --> 00:29:43,900 And that's what we mean by the real exponential. 432 00:29:43,900 --> 00:29:47,010 Shortly, we'll also consider complex exponentials, where 433 00:29:47,010 --> 00:29:50,240 these numbers can then become complex. 434 00:29:50,240 --> 00:29:54,120 So this is an exponential function. 435 00:29:54,120 --> 00:29:59,220 And for example, if the parameter a is positive, that 436 00:29:59,220 --> 00:30:03,320 means that we have a growing exponential function. 437 00:30:03,320 --> 00:30:08,410 If the parameter a is negative, then that means that 438 00:30:08,410 --> 00:30:13,760 we have a decaying exponential function. 439 00:30:13,760 --> 00:30:17,750 Now, somewhat as an aside, it's kind of interesting to 440 00:30:17,750 --> 00:30:23,160 note that for exponentials, a time shift corresponds to a 441 00:30:23,160 --> 00:30:26,950 scale change, which is somewhat different than what 442 00:30:26,950 --> 00:30:29,700 happens with sinusoids. 443 00:30:29,700 --> 00:30:33,380 In the sinusoidal case, we saw that a time shift corresponded 444 00:30:33,380 --> 00:30:34,940 to a phase change. 445 00:30:34,940 --> 00:30:37,850 With the real exponential, a time shift, as it turns out, 446 00:30:37,850 --> 00:30:41,070 corresponds to simply changing the scale. 447 00:30:41,070 --> 00:30:45,730 There's nothing particularly crucial or 448 00:30:45,730 --> 00:30:47,300 exciting about that. 449 00:30:47,300 --> 00:30:51,050 And in fact, perhaps stressing it is a little misleading. 450 00:30:51,050 --> 00:30:53,620 For general functions, of course, about all that you can 451 00:30:53,620 --> 00:30:58,430 say about what happens when you implement a time shift is 452 00:30:58,430 --> 00:31:02,720 that it implements a time shift. 453 00:31:02,720 --> 00:31:03,060 OK. 454 00:31:03,060 --> 00:31:05,890 So here's the real exponential. 455 00:31:05,890 --> 00:31:08,600 Just C e ^ (a t). 456 00:31:08,600 --> 00:31:12,160 Let's look at the real exponential, now, in the 457 00:31:12,160 --> 00:31:14,480 discrete-time case. 458 00:31:14,480 --> 00:31:21,550 And in the discrete-time case, we have several alternate ways 459 00:31:21,550 --> 00:31:23,910 of expressing it. 460 00:31:23,910 --> 00:31:29,020 We can express the real exponential in the form C e ^ 461 00:31:29,020 --> 00:31:35,120 (beta n), or as we'll find more convenient, in part for a 462 00:31:35,120 --> 00:31:38,810 reason at I'll indicate shortly, we can rewrite this 463 00:31:38,810 --> 00:31:44,820 as C alpha ^ n, where of course, alpha = e ^ beta. 464 00:31:44,820 --> 00:31:47,610 More typically in the discrete-time case, we'll 465 00:31:47,610 --> 00:31:53,720 express the exponential as C alpha ^ n. 466 00:31:53,720 --> 00:31:57,690 So for example, this becomes, essentially, a geometric 467 00:31:57,690 --> 00:32:01,920 series or progression as n continues for 468 00:32:01,920 --> 00:32:03,740 certain values of alpha. 469 00:32:03,740 --> 00:32:11,070 Here for example, we have for alpha greater than 0, first of 470 00:32:11,070 --> 00:32:15,010 all on the top, the case where the magnitude of alpha is 471 00:32:15,010 --> 00:32:21,430 greater than 1, so that the sequence is exponentially or 472 00:32:21,430 --> 00:32:23,420 geometrically growing. 473 00:32:23,420 --> 00:32:27,770 On the bottom, again with alpha positive, but now with 474 00:32:27,770 --> 00:32:32,517 its magnitude less than 1, we have a geometric progression 475 00:32:32,517 --> 00:32:38,360 that is exponentially or geometrically decaying. 476 00:32:38,360 --> 00:32:38,650 OK. 477 00:32:38,650 --> 00:32:42,350 So this, in both of these cases, is with alpha 478 00:32:42,350 --> 00:32:45,070 greater than 0. 479 00:32:45,070 --> 00:32:48,480 Now the function that we're talking about is alpha ^ n. 480 00:32:48,480 --> 00:32:51,310 And of course, what you can see is that if alpha is 481 00:32:51,310 --> 00:32:57,910 negative instead of positive, then when n is even, that 482 00:32:57,910 --> 00:32:59,620 minus sign is going to disappear. 483 00:32:59,620 --> 00:33:02,320 When n is odd, there will be a minus sign. 484 00:33:02,320 --> 00:33:06,610 And so for alpha negative, the sequence is going to alternate 485 00:33:06,610 --> 00:33:08,500 positive and negative values. 486 00:33:08,500 --> 00:33:15,870 So for example, here we have alpha negative, with its 487 00:33:15,870 --> 00:33:17,230 magnitude less than 1. 488 00:33:17,230 --> 00:33:21,130 And you can see that, again, its envelope decays 489 00:33:21,130 --> 00:33:25,320 geometrically, and the values alternate in sign. 490 00:33:25,320 --> 00:33:29,310 And here we have the magnitude of alpha greater than 1, with 491 00:33:29,310 --> 00:33:30,600 alpha negative. 492 00:33:30,600 --> 00:33:33,360 Again, they alternate in sign, and of course it's growing 493 00:33:33,360 --> 00:33:35,740 geometrically. 494 00:33:35,740 --> 00:33:42,910 Now, if you think about alpha positive and go back to the 495 00:33:42,910 --> 00:33:48,640 expression that I have at the top, namely C alpha ^ n. 496 00:33:48,640 --> 00:33:54,670 With alpha positive, you can see a straightforward 497 00:33:54,670 --> 00:33:58,440 relationship between alpha and beta. 498 00:33:58,440 --> 00:34:01,830 Namely, beta is the natural logarithm of alpha. 499 00:34:04,520 --> 00:34:06,830 Something to think about is what 500 00:34:06,830 --> 00:34:09,070 happens if alpha is negative? 501 00:34:09,070 --> 00:34:13,230 Which is, of course, a very important and useful class of 502 00:34:13,230 --> 00:34:16,190 real discrete-time exponentials also. 503 00:34:16,190 --> 00:34:19,560 Well, it turns out that with alpha negative, if you try to 504 00:34:19,560 --> 00:34:25,020 express it as C e ^ (beta n), then beta comes out to be an 505 00:34:25,020 --> 00:34:26,659 imaginary number. 506 00:34:26,659 --> 00:34:30,889 And that is one, but not the only reason why, in the 507 00:34:30,889 --> 00:34:36,389 discrete-time case, it's often most convenient to phrase real 508 00:34:36,389 --> 00:34:42,420 exponentials in the form alpha ^ n, rather than e ^ (beta n). 509 00:34:42,420 --> 00:34:47,650 In other words, to express them in this form rather than 510 00:34:47,650 --> 00:34:48,900 in this form. 511 00:34:54,520 --> 00:34:56,699 Those are real exponentials, continuous-time and 512 00:34:56,699 --> 00:34:57,960 discrete-time. 513 00:34:57,960 --> 00:35:01,710 Now let's look at the continuous-time complex 514 00:35:01,710 --> 00:35:03,130 exponential. 515 00:35:03,130 --> 00:35:09,910 And what I mean by a complex exponential, again, is an 516 00:35:09,910 --> 00:35:15,750 exponential of the form C e ^ (a t). 517 00:35:15,750 --> 00:35:21,050 But in this case, we allow the parameters C and a to be 518 00:35:21,050 --> 00:35:22,970 complex numbers. 519 00:35:22,970 --> 00:35:25,850 And let's just track this through algebraically. 520 00:35:25,850 --> 00:35:28,820 If C and a are complex numbers, let's write C in 521 00:35:28,820 --> 00:35:33,350 polar form, so it has a magnitude and an angle. 522 00:35:33,350 --> 00:35:36,800 Let's write a in rectangular form, so it has a real part 523 00:35:36,800 --> 00:35:38,950 and an imaginary part. 524 00:35:38,950 --> 00:35:43,060 And when we substitute these two in here, combine some 525 00:35:43,060 --> 00:35:44,680 things together-- 526 00:35:44,680 --> 00:35:46,100 well actually, I haven't combined yet. 527 00:35:46,100 --> 00:35:52,160 I have this for the amplitude factor, and this for the 528 00:35:52,160 --> 00:35:53,960 exponential factor. 529 00:35:53,960 --> 00:35:58,597 I can now pull out of this the term corresponding to e ^ (r 530 00:35:58,597 --> 00:36:02,320 t), and combine the imaginary parts together. 531 00:36:02,320 --> 00:36:05,760 And I come down to the expression that I have here. 532 00:36:05,760 --> 00:36:12,800 So following this further, an exponential of this form, e ^ 533 00:36:12,800 --> 00:36:19,640 (j omega) or e ^ (j phi), using Euler's relation, can be 534 00:36:19,640 --> 00:36:28,150 expressed as the sum of a cosine plus j times a sine. 535 00:36:28,150 --> 00:36:31,560 And so that corresponds to this factor. 536 00:36:31,560 --> 00:36:35,290 And then there is this time-varying amplitude factor 537 00:36:35,290 --> 00:36:36,420 on top of it. 538 00:36:36,420 --> 00:36:39,390 Finally putting those together, we end up with the 539 00:36:39,390 --> 00:36:42,580 expression that I show on the bottom. 540 00:36:42,580 --> 00:36:49,010 And what this corresponds to are two sinusoidal signals, 90 541 00:36:49,010 --> 00:36:51,733 degrees out of phase, as indicated by the fact that 542 00:36:51,733 --> 00:36:54,220 there's a cosine and a sine. 543 00:36:54,220 --> 00:36:57,560 So there's a real part and an imaginary part, with 544 00:36:57,560 --> 00:37:03,210 sinusoidal components 90 degrees out of phase, and a 545 00:37:03,210 --> 00:37:07,140 time-varying amplitude factor, which is a real exponential. 546 00:37:07,140 --> 00:37:11,670 So it's a sinusoid multiplied by a real exponential in both 547 00:37:11,670 --> 00:37:14,870 the real part and the imaginary part. 548 00:37:14,870 --> 00:37:19,070 And let's just see what one of those terms might look like. 549 00:37:19,070 --> 00:37:29,290 What I've indicated at the top is a sinusoidal signal with a 550 00:37:29,290 --> 00:37:32,330 time-varying exponential envelope, or an envelope which 551 00:37:32,330 --> 00:37:37,780 is a real exponential, and in particular which is growing, 552 00:37:37,780 --> 00:37:40,960 namely with r greater than 0. 553 00:37:40,960 --> 00:37:44,430 And on the bottom, I've indicated the same thing with 554 00:37:44,430 --> 00:37:46,240 r less than 0. 555 00:37:46,240 --> 00:37:51,300 And this kind of sinusoidal signal, by the way, is 556 00:37:51,300 --> 00:37:54,730 typically referred to as a damped sinusoid. 557 00:37:54,730 --> 00:37:58,110 So with r negative, what we have in the real and imaginary 558 00:37:58,110 --> 00:38:01,130 parts are damped sinusoids. 559 00:38:01,130 --> 00:38:07,270 And the sinusoidal components of that are 90 degrees out of 560 00:38:07,270 --> 00:38:12,260 phase, in the real part and in the imaginary part. 561 00:38:12,260 --> 00:38:12,560 OK. 562 00:38:12,560 --> 00:38:17,040 Now, in the discrete-time case, we have more or less the 563 00:38:17,040 --> 00:38:19,350 same kind of outcome. 564 00:38:19,350 --> 00:38:26,220 In particular we'll make reference to our complex 565 00:38:26,220 --> 00:38:28,860 exponentials in the discrete-time case. 566 00:38:28,860 --> 00:38:33,200 The expression for the complex exponential looks very much 567 00:38:33,200 --> 00:38:36,430 like the expression for the real exponential, except that 568 00:38:36,430 --> 00:38:40,460 now we have complex factors. 569 00:38:40,460 --> 00:38:46,040 So C and alpha are complex numbers. 570 00:38:46,040 --> 00:38:53,050 And again, if we track through the algebra, and get to a 571 00:38:53,050 --> 00:38:57,740 point where we have a real exponential multiplied by a 572 00:38:57,740 --> 00:39:03,930 factor which is a purely imaginary exponential, apply 573 00:39:03,930 --> 00:39:10,430 Euler's relationship to this, we then finally come down to a 574 00:39:10,430 --> 00:39:16,980 sequence, which has a real exponential amplitude 575 00:39:16,980 --> 00:39:21,480 multiplying one sinusoid in the real part. 576 00:39:21,480 --> 00:39:25,120 And in the imaginary part, exactly the same kind of 577 00:39:25,120 --> 00:39:30,560 exponential multiplying a sinusoid that's 90 degrees out 578 00:39:30,560 --> 00:39:33,150 of phase from that. 579 00:39:33,150 --> 00:39:37,750 And so if we look at what one of these factors might look 580 00:39:37,750 --> 00:39:42,510 like, it's what we would expect given the analogy with 581 00:39:42,510 --> 00:39:44,580 the continuous-time case. 582 00:39:44,580 --> 00:39:53,140 Namely, it's a sinusoidal sequence with a real 583 00:39:53,140 --> 00:39:55,060 exponential envelope. 584 00:39:55,060 --> 00:39:58,500 In the case where alpha is positive, then 585 00:39:58,500 --> 00:40:00,250 it's a growing envelope. 586 00:40:00,250 --> 00:40:03,050 In the case where alpha is negative-- 587 00:40:03,050 --> 00:40:03,990 I'm sorry-- 588 00:40:03,990 --> 00:40:08,200 where the magnitude of alpha is greater than 1, it's a 589 00:40:08,200 --> 00:40:10,070 growing exponential envelope. 590 00:40:10,070 --> 00:40:14,700 Where the magnitude of alpha is less than 1, it's a 591 00:40:14,700 --> 00:40:17,570 decaying exponential envelope. 592 00:40:17,570 --> 00:40:20,370 And so I've illustrated that here. 593 00:40:20,370 --> 00:40:24,050 Here we have the magnitude of alpha greater than 1. 594 00:40:24,050 --> 00:40:27,510 And here we have the magnitude of alpha less than 1. 595 00:40:27,510 --> 00:40:30,960 In both cases, sinusoidal sequences underneath the 596 00:40:30,960 --> 00:40:35,920 envelope, and then an envelope that is dictated by what the 597 00:40:35,920 --> 00:40:39,710 magnitude of alpha is. 598 00:40:39,710 --> 00:40:40,200 OK. 599 00:40:40,200 --> 00:40:44,900 Now, in the discrete-time case, then, we have results 600 00:40:44,900 --> 00:40:47,080 similar to the continuous-time case. 601 00:40:47,080 --> 00:40:53,170 Namely, components in a real and imaginary part that have a 602 00:40:53,170 --> 00:40:57,370 real exponential factor times a sinusoid. 603 00:40:57,370 --> 00:41:07,240 Of course, if the magnitude of alpha is equal to 1, then this 604 00:41:07,240 --> 00:41:10,010 factor disappears, or is equal to 1. 605 00:41:10,010 --> 00:41:12,350 And this factor is equal to 1. 606 00:41:12,350 --> 00:41:15,250 And so we have sinusoids in both the real 607 00:41:15,250 --> 00:41:17,620 and imaginary parts. 608 00:41:17,620 --> 00:41:22,660 Now, one can ask whether, in general, the complex 609 00:41:22,660 --> 00:41:26,580 exponential with the magnitude of alpha equal to 1 is 610 00:41:26,580 --> 00:41:28,640 periodic or not periodic. 611 00:41:28,640 --> 00:41:34,450 And the clue to that can be inferred by examining this 612 00:41:34,450 --> 00:41:36,680 expression. 613 00:41:36,680 --> 00:41:39,600 In particular, in the discrete-time case with the 614 00:41:39,600 --> 00:41:44,600 magnitude of alpha equal to 1, we have pure sinusoids in the 615 00:41:44,600 --> 00:41:48,250 real part and the imaginary part. 616 00:41:48,250 --> 00:41:51,240 And in fact, in a continuous-time case with r 617 00:41:51,240 --> 00:41:56,910 equal to 0, we have sinusoids in the real part and the 618 00:41:56,910 --> 00:41:59,570 imaginary part. 619 00:41:59,570 --> 00:42:02,520 In a continuous-time case when we have a pure complex 620 00:42:02,520 --> 00:42:07,580 exponential, so that the terms aren't exponentially growing 621 00:42:07,580 --> 00:42:09,410 or decaying, those 622 00:42:09,410 --> 00:42:13,420 exponentials are always periodic. 623 00:42:13,420 --> 00:42:17,630 Because, of course, the real and imaginary sinusoidal 624 00:42:17,630 --> 00:42:19,640 components are periodic. 625 00:42:19,640 --> 00:42:23,100 In the discrete-time case, we know that the sinusoids may or 626 00:42:23,100 --> 00:42:28,150 may not be periodic, depending on the value of omega_0. 627 00:42:28,150 --> 00:42:31,390 And so in fact, in the discrete-time case, the 628 00:42:31,390 --> 00:42:38,630 exponential e ^ (j omega_0 n), that I've indicated here, may 629 00:42:38,630 --> 00:42:42,780 or may not be periodic depending on what the value of 630 00:42:42,780 --> 00:42:44,030 omega_0 is. 631 00:42:47,140 --> 00:42:47,660 OK. 632 00:42:47,660 --> 00:42:53,410 Now, to summarize, in this lecture I've introduced and 633 00:42:53,410 --> 00:42:58,000 discussed a number of important basic signals. 634 00:42:58,000 --> 00:43:04,710 In particular, sinusoids and real and complex exponentials. 635 00:43:04,710 --> 00:43:07,660 One of the important outcomes of the discussion, emphasized 636 00:43:07,660 --> 00:43:11,190 further in the text, is that there are some very important 637 00:43:11,190 --> 00:43:13,060 similarities between them. 638 00:43:13,060 --> 00:43:16,700 But there are also some very important differences. 639 00:43:16,700 --> 00:43:20,770 And these differences will surface when we exploit 640 00:43:20,770 --> 00:43:26,220 sinusoids and complex exponentials as basic building 641 00:43:26,220 --> 00:43:29,350 blocks for more general continuous-time and 642 00:43:29,350 --> 00:43:32,160 discrete-time signals. 643 00:43:32,160 --> 00:43:35,900 In the next lecture, what I'll discuss are some other very 644 00:43:35,900 --> 00:43:40,070 important building blocks, namely, what are referred to 645 00:43:40,070 --> 00:43:43,230 as step signals and impulse signals. 646 00:43:43,230 --> 00:43:47,970 And those, together with the sinusoidal signals and 647 00:43:47,970 --> 00:43:52,160 exponentials as we've talked about today, will really form 648 00:43:52,160 --> 00:43:57,300 the cornerstone for, essentially, all of the signal 649 00:43:57,300 --> 00:44:00,270 and system analysis that we'll be dealing with for the 650 00:44:00,270 --> 00:44:01,780 remainder of course. 651 00:44:01,780 --> 00:44:03,030 Thank you.